Integrand size = 18, antiderivative size = 36 \[ \int \frac {\sqrt {-1+x}}{x^2 \sqrt {1+x}} \, dx=-\frac {\sqrt {-1+x} \sqrt {1+x}}{x}+\arctan \left (\sqrt {-1+x} \sqrt {1+x}\right ) \]
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Time = 0.00 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {96, 94, 209} \[ \int \frac {\sqrt {-1+x}}{x^2 \sqrt {1+x}} \, dx=\arctan \left (\sqrt {x-1} \sqrt {x+1}\right )-\frac {\sqrt {x-1} \sqrt {x+1}}{x} \]
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Rule 94
Rule 96
Rule 209
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {-1+x} \sqrt {1+x}}{x}+\int \frac {1}{\sqrt {-1+x} x \sqrt {1+x}} \, dx \\ & = -\frac {\sqrt {-1+x} \sqrt {1+x}}{x}+\text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {-1+x} \sqrt {1+x}\right ) \\ & = -\frac {\sqrt {-1+x} \sqrt {1+x}}{x}+\tan ^{-1}\left (\sqrt {-1+x} \sqrt {1+x}\right ) \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.83 \[ \int \frac {\sqrt {-1+x}}{x^2 \sqrt {1+x}} \, dx=-\frac {\sqrt {\frac {-1+x}{1+x}} \left (\sqrt {-1+x} (1+x)+2 x \sqrt {1+x} \arctan \left (x-\sqrt {-1+x} \sqrt {1+x}\right )\right )}{\sqrt {-1+x} x} \]
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Time = 1.05 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.19
method | result | size |
default | \(\frac {\left (-\arctan \left (\frac {1}{\sqrt {x^{2}-1}}\right ) x -\sqrt {x^{2}-1}\right ) \sqrt {x -1}\, \sqrt {x +1}}{x \sqrt {x^{2}-1}}\) | \(43\) |
risch | \(-\frac {\sqrt {x -1}\, \sqrt {x +1}}{x}-\frac {\arctan \left (\frac {1}{\sqrt {x^{2}-1}}\right ) \sqrt {\left (x -1\right ) \left (x +1\right )}}{\sqrt {x -1}\, \sqrt {x +1}}\) | \(46\) |
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Time = 0.26 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.08 \[ \int \frac {\sqrt {-1+x}}{x^2 \sqrt {1+x}} \, dx=\frac {2 \, x \arctan \left (\sqrt {x + 1} \sqrt {x - 1} - x\right ) - \sqrt {x + 1} \sqrt {x - 1} - x}{x} \]
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\[ \int \frac {\sqrt {-1+x}}{x^2 \sqrt {1+x}} \, dx=\int \frac {\sqrt {x - 1}}{x^{2} \sqrt {x + 1}}\, dx \]
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Time = 0.27 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.56 \[ \int \frac {\sqrt {-1+x}}{x^2 \sqrt {1+x}} \, dx=-\frac {\sqrt {x^{2} - 1}}{x} - \arcsin \left (\frac {1}{{\left | x \right |}}\right ) \]
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none
Time = 0.32 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.17 \[ \int \frac {\sqrt {-1+x}}{x^2 \sqrt {1+x}} \, dx=-\frac {8}{{\left (\sqrt {x + 1} - \sqrt {x - 1}\right )}^{4} + 4} - 2 \, \arctan \left (\frac {1}{2} \, {\left (\sqrt {x + 1} - \sqrt {x - 1}\right )}^{2}\right ) \]
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Time = 19.79 (sec) , antiderivative size = 138, normalized size of antiderivative = 3.83 \[ \int \frac {\sqrt {-1+x}}{x^2 \sqrt {1+x}} \, dx=-\ln \left (\frac {{\left (\sqrt {x-1}-\mathrm {i}\right )}^2}{{\left (\sqrt {x+1}-1\right )}^2}+1\right )\,1{}\mathrm {i}+\ln \left (\frac {\sqrt {x-1}-\mathrm {i}}{\sqrt {x+1}-1}\right )\,1{}\mathrm {i}-\frac {\sqrt {x-1}-\mathrm {i}}{4\,\left (\sqrt {x+1}-1\right )}-\frac {\frac {5\,{\left (\sqrt {x-1}-\mathrm {i}\right )}^2}{{\left (\sqrt {x+1}-1\right )}^2}+1}{\frac {4\,{\left (\sqrt {x-1}-\mathrm {i}\right )}^3}{{\left (\sqrt {x+1}-1\right )}^3}+\frac {4\,\left (\sqrt {x-1}-\mathrm {i}\right )}{\sqrt {x+1}-1}} \]
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